↑

Properties

Label 20.10.9892112435...0000.1
Degree $20$
Signature $[10, 5]$
Discriminant $-\,2^{8}\cdot 5^{10}\cdot 13^{12}\cdot 19^{8}$
Root discriminant $44.64$
Ramified primes $2, 5, 13, 19$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T633

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 0, -24, 0, 50, 0, 590, 0, -18, 0, -2064, 0, -848, 0, 806, 0, -102, 0, -6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^18 - 102*x^16 + 806*x^14 - 848*x^12 - 2064*x^10 - 18*x^8 + 590*x^6 + 50*x^4 - 24*x^2 - 1)
 
gp: K = bnfinit(x^20 - 6*x^18 - 102*x^16 + 806*x^14 - 848*x^12 - 2064*x^10 - 18*x^8 + 590*x^6 + 50*x^4 - 24*x^2 - 1, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{18} - 102 x^{16} + 806 x^{14} - 848 x^{12} - 2064 x^{10} - 18 x^{8} + 590 x^{6} + 50 x^{4} - 24 x^{2} - 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-989211243530600674561802500000000=-\,2^{8}\cdot 5^{10}\cdot 13^{12}\cdot 19^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.64$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 5, 13, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{10} - \frac{1}{2} a^{7} - \frac{1}{4} a^{5} - \frac{1}{2} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{11} - \frac{1}{2} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{3} + \frac{1}{4} a$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{12} - \frac{1}{2} a^{8} + \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{1216605820196} a^{18} - \frac{49853639513}{608302910098} a^{16} + \frac{53785258823}{608302910098} a^{14} - \frac{1}{4} a^{13} - \frac{21247534925}{608302910098} a^{12} - \frac{75874854871}{304151455049} a^{10} - \frac{47121162061}{1216605820196} a^{8} + \frac{35910401311}{608302910098} a^{6} - \frac{1}{2} a^{5} - \frac{115368005591}{608302910098} a^{4} + \frac{1}{4} a^{3} + \frac{59539663727}{608302910098} a^{2} - \frac{32421839089}{608302910098}$, $\frac{1}{2433211640392} a^{19} - \frac{1}{2433211640392} a^{18} + \frac{204444176023}{2433211640392} a^{17} - \frac{204444176023}{2433211640392} a^{16} - \frac{196580937403}{2433211640392} a^{15} + \frac{196580937403}{2433211640392} a^{14} + \frac{261656385199}{2433211640392} a^{13} - \frac{261656385199}{2433211640392} a^{12} - \frac{607650874533}{2433211640392} a^{11} + \frac{607650874533}{2433211640392} a^{10} - \frac{47121162061}{2433211640392} a^{9} + \frac{47121162061}{2433211640392} a^{8} - \frac{232330652427}{2433211640392} a^{7} + \frac{232330652427}{2433211640392} a^{6} - \frac{1143190376329}{2433211640392} a^{5} + \frac{1143190376329}{2433211640392} a^{4} - \frac{185072127595}{2433211640392} a^{3} + \frac{185072127595}{2433211640392} a^{2} + \frac{847610686969}{2433211640392} a - \frac{847610686969}{2433211640392}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $14$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 2704412724.43 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T633:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40960
The 124 conjugacy class representatives for t20n633 are not computed
Character table for t20n633 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.19827925.1, 10.10.1965733049028125.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 20 siblings: data not computed
Degree 40 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R $20$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ R $20$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.8.0.1$x^{8} + x^{4} + x^{3} + x + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
2.8.8.8$x^{8} + 4 x^{5} + 8 x^{2} + 48$$2$$4$$8$$((C_8 : C_2):C_2):C_2$$[2, 2, 2, 2]^{4}$
5Data not computed
13Data not computed
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.2.2$x^{4} - 19 x^{2} + 722$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
19.4.2.2$x^{4} - 19 x^{2} + 722$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
19.8.4.1$x^{8} + 7220 x^{4} - 27436 x^{2} + 13032100$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$